Defining polynomial
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\(x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7\)
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Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $6$ |
| Ramification exponent $e$: | $3$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $4$ |
| Discriminant root field: | $\Q_{2}(\sqrt{5})$ |
| Root number: | $1$ |
| $\card{ \Gal(K/\Q_{ 2 }) }$: | $6$ |
| This field is Galois over $\Q_{2}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{2}(\sqrt{5})$, 2.3.2.1 x3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{2}(\sqrt{5})$ $\cong \Q_{2}(t)$ where $t$ is a root of
\( x^{2} + x + 1 \)
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| Relative Eisenstein polynomial: |
\( x^{3} + 2 x + 2 \)
$\ \in\Q_{2}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z^{2} + z + 1$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois group: | $S_3$ (as 6T2) |
| Inertia group: | Intransitive group isomorphic to $C_3$ |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $2$ |
| Tame degree: | $3$ |
| Wild slopes: | None |
| Galois mean slope: | $2/3$ |
| Galois splitting model: |
$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$
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